The Problem with Cronbach's Alpha: Comment on Sijtsma and ...
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University of Pennsylvania University of Pennsylvania
ScholarlyCommons ScholarlyCommons
School of Nursing Departmental Papers School of Nursing
3-2015
The Problem with Cronbach's Alpha: Comment on Sijtsma and The Problem with Cronbach's Alpha: Comment on Sijtsma and
Van der Ark (2015) Van der Ark (2015)
Claudio Barbaranelli
Christopher S. Lee
Ercole Vellone
Barbara Riegel University of Pennsylvania, [email protected]
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Recommended Citation Recommended Citation Barbaranelli, C., Lee, C. S., Vellone, E., & Riegel, B. (2015). The Problem with Cronbach's Alpha: Comment on Sijtsma and Van der Ark (2015). Nursing Research, 64 (2), 140-145. http://dx.doi.org/10.1097/NNR.0000000000000079
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/nrs/105 For more information, please contact [email protected].
The Problem with Cronbach's Alpha: Comment on Sijtsma and Van der Ark The Problem with Cronbach's Alpha: Comment on Sijtsma and Van der Ark (2015) (2015)
Abstract Abstract Knowledge of a scale's dimensionality is an essential preliminary step to the application of any measure of reliability derived from classical test theory--an approach commonly used is nursing research. The focus of this article is on the applied aspects of reliability and dimensionality testing. Throughout the article, the Self-Care of Heart Failure Index is used to exemplify real-world data challenges of quantifying reliability and to provide insight into how to overcome such challenges.
Keywords Keywords Humans, Models, Statistical, Nursing Research, Reproducibility of Results
Disciplines Disciplines Analytical, Diagnostic and Therapeutic Techniques and Equipment | Cardiology | Cardiovascular Diseases | Circulatory and Respiratory Physiology | Health and Medical Administration | Health Services Administration | Health Services Research | Medical Humanities | Medicine and Health Sciences | Nursing | Preventive Medicine
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/nrs/105
The Problem with Cronbach's Alpha: Comment on Sijtsma and Van der Ark (2015)
Claudio Barbaranelli, PhD [Professor],Department of Psychology Sapienza University of Rome, Italy
Christopher S. Lee, PhD, RN [Associate Professor],Schools of Nursing and Medicine, Oregon Health & Science University, Portland, OR
Ercole Vellone, PhD, RN [Research Fellow], andDepartment of Biomedicine and Prevention Tor Vergata University, Rome, Italy
Barbara Riegel, DNSc, RN, FAAN, FAHA [Professor]School of Nursing University of Pennsylvania, Philadelphia, PA
Abstract
Knowledge of a scale's dimensionality is an essential preliminary step to the application of any
measure of reliability derived from classical test theory—an approach commonly used is nursing
research. The focus of this article is on the applied aspects of reliability and dimensionality testing.
Throughout the article, the Self-Care of Heart Failure Index is used to exemplify real-world data
challenges of quantifying reliability, and to provide insight into how to overcome such challenges.
Keywords
psychological measurement; psychometrics; questionnaires; reliability; reproducibility of results; Self-Care of Heart Failure Index
Sijtsma and Van der Ark discussed three approaches to reliability: classical test theory
(CTT); factor analysis (FA); and generalizability theory (GT) in the special and infrequent
case of three-way data structure (i.e., when there is more than one rater). It is much more
often the case, however, that scientists in nursing and other health disciplines are concerned
with the reliability of a measure that is ascertained from a single respondent such as a study
participant. Sijtsma and Van der Ark correctly identified that Sirotnik (1970) demonstrated
that the generalizability coefficient derived from GT-based approaches is equivalent to
coefficient alpha when there is a single respondent to the items of interest. Thus, GT-based
methods provide no additional value over CTT and FA approaches for the most pressing and
real-world challenges in reliability that we face. Moreover, the examination of reliability
according to CTT principles and examination of scale dimensionality (through FA) are
oftentimes undertaken as if they were separate issues. We believe that knowledge of a scale's
Corresponding Author: Barbara Riegel, DNSc, RN, FAAN, FAHA, Professor, School of Nursing, University of Pennsylvania,418 Curie Boulevard, Philadelphia, PA19104-4217 ([email protected]).
The authors have no conflicts of interest to report.
HHS Public AccessAuthor manuscriptNurs Res. Author manuscript; available in PMC 2016 March 01.
Published in final edited form as:Nurs Res. 2015 ; 64(2): 140–145. doi:10.1097/NNR.0000000000000079.
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dimensionality is an essential preliminary step to the application of any measure of
reliability developed within the framework of CTT.
Accordingly, this article focuses on the applied aspects of reliability and dimensionality
testing. As GT applies only to designs uncommon in nursing research, we focus specifically
on CTT and FA, which are commonly used in nursing research. Throughout the article, we
use the Self-Care of Heart Failure Index (SCHFI; Riegel, Lee, Dickson, & Carlson, 2009) to
exemplify real-world data challenges of quantifying reliability and to provide insight into
how to overcome such challenges.
Reliability Assessment and the Study of Scale Dimensionality
A first issue raised by Sijtsma and Van der Ark that we want to address is related to the
relationship between CTT and FA models. We believe that it is crucial to first examine scale
dimensionality and then conduct FA before quantifying any index of reliability. This advice
applies to Cronbach's alpha in particular because this approach assumes a unidimensional
scale.
As noted by Sijtsma and Van der Ark, in CTT there is no assumption of covariance among
items that have been produced to measure a construct within a scale. As such, there is no
reason to expect that a scale's items would be correlated because CTT models simply
decompose measurement values into systematic and random measurement parts (Sijtsma
and Van der Ark). For the hypothetical nature of a CTT model, however, reliability cannot
be computed directly from the basic formulas that define the model. Instead, reliability
needs to be estimated by other models (see Sijtsma and Van der Ark, p. xx). A common
strategy to get these estimates is based on the use of scores from a single-test, single-
administration design, where “estimates based on the covariances of all item pairs provide
lower bounds to the reliability” (Sijtsma and Van der Ark, p. 11), and Cronbach's alpha “is
the best known representative of this category” (Sijtsma and Van der Ark, p. 11). As is well
known, a scale's alpha is a function of correlations among scale items, and other things
being equal, the more substantial the average items correlate the higher the alpha. Then,
while the “theoretical” CTT model does not require that items are correlated in order to have
a reliable measure, the more commonly used method for deriving reliability estimates
provides better estimates when items are highly correlated (other things being equal).
The dependence of CTT-based reliability on more common methods for estimating
reliability within the same model stems from the fact that CTT is a hypothetical model that
is not anchored in reality. As specified by Sijtsma and Van der Ark, the theoretical model in
CTT implies the administration of the same items over and over again to the same
individual. In contrast, reality implies that a set of items is administered once or maybe
twice to a sample of individuals. In the extension of this hypothetical model to reality, it is
necessary that items composing a scale at least measure the same construct—if not with the
same precision—in order to obtain unbiased measures of reliability by means of alpha
coefficient (i.e. the tau-equivalence or true score equivalence; Raykov & Marcoulides,
2011). We are now faced with a paradox. Specifically, the CTT definition of reliability
ignores what each item measures, and items are not required to be correlated. But in the real
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world, the method used for estimating reliability with this model cannot ignore what items
measure; alpha is an accurate measure of reliability only if items measure the same construct
and are sufficiently correlated.
In the practical application of CTT, one cannot ignore that the items are correlated. In fact,
alpha must be high to empirically demonstrate reliability. That is, items must measure the
same construct with the same unit of measurement in order to have an unbiased measure of
reliability. If one considers the formula for Cronbach's alpha, however, it is clear that alpha
depends only on item average correlations (or inter-item covariances)(Sijtsma, 2009),
assuming they are all positive and from the number of items forming the scale (Nunnally &
Bernstein, 1994). In contrast, alpha is not dependent on a particular structure. In fact, item
intercorrelations may be generated by different latent structures. Here is where FA comes
into play. Factor analysis does not simply assume that items are correlated, but FA gives a
structure to the correlations in order to explain their origin.
Confusing homogeneity with internal consistency, high values of Cronbach's alpha are often
improperly interpreted as evidence of a single factor explaining correlations among scale
items (Raykov, 2012; Raykov & Marcoulides, 2011). Homogeneity refers to a situation
when there is only one latent dimension explaining item correlations, while internal
consistency refers to the intercorrelation among scale items (Schmitt, 1996). Since alpha is
an index of internal consistency, it provides no information regarding the number of factors
explaining item correlations. Such information can be obtained only after a careful
examination of the items’ latent structure (e.g., dimensionality)—obtained from FA. Simply
put, only the results of FA provide evidence of which items should be summed in order to
obtain scores that reflect the intended construct. Relying solely on the results of alpha—to
inform structure—poses a risk of ending up with summed scores that reflect mixed sources
of variance.
Due to the great interdependency of reliability on dimensionality, we argue that scale
dimensionality must be assessed with FA before choosing an appropriate method of
estimating reliability. A variety of reliability indices have been developed. Below we
describe how the results of FA can be used to identify which reliability index to use.
Some Reliability Indices from Which to Choose After Assessing
Dimensionality
When items are congeneric (i.e., a model with one homogenous factor that fits the data) and
measurement errors are not correlated, composite scale reliability can be defined as shown
in Equation 1 (Fornell & Larcker, 1981):
(1)
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where λ is the factor loading of the indicator, and Var(εy) is its residual variance—assuming
that φ, the variance of the factor—is fixed at 1 for identification purposes, and that the factor
and the measurement error are independent. This coefficient gives consistent estimators of
scale reliability (Raykov, 2012), and has a close resemblance to the omega coefficient
discussed below (McDonald, 1999; Revelle & Zinbarg, 2009).
While omega can be used as a measure of reliability when one latent variable accounts for
item correlations (McDonald, 1999), the composite scale reliability coefficient can be
extended to consider the more general case of a multidimensional scale, that is, a scale
where more than one latent variable explains correlations among observed variables in a
dataset (Camilli, Wang, & Fesq, 1995).
The formula for a global reliability index for multidimensional scales proposed by Raykov
and Marcoulides (2011) is shown in Equation 2:
(2)
where A is the matrix of factor loadings onto common factors and A’ its transpose, Φ is the
covariance matrix of the common factors, Θ is the covariance matrix of measurement errors,
1 is a unity vector and 1’ its transpose.
Another index that can be used when a scale has more than one factor is the model-based
internal consistency index proposed by Bentler (2009) and shown in Equation 3:
(3)
where is the fitted covariance matrix obtained from model parameter estimates, and is
the error covariance matrix.
All these coefficients can be easily computed using standard output from structural equation
modeling programs, as well as standard output from software for exploratory FA. We do not
have space here to dwell on a discussion of other indices. The interested reader is referred to
Raykov (2012) and Revelle and Zinbarg (2009).
Empirical Examples: The Self-Care of Health Failure (SCHFI)
As noted in the introduction, the empirical examples we provide refer to the analysis of the
SCHFI (Riegel et al., 2009)—a measure that has broad use in clinical research and practice,
in spite of the fact that we have been grappling for years with its reliability. The situation-
specific theory of heart failure (HF) self-care (Riegel & Dickson, 2008) defines self-care as
a naturalistic decision-making process composed of two dimensions: self-care maintenance
and self-care management. Self-care maintenance reflects behaviors in which patients
engage to maintain stability and prevent an exacerbation, including monitoring for signs and
symptoms like dyspnea and edema, and adhering to prescribed therapies like medicines and
dietary restrictions. Self-care management is a process of recognizing symptoms when they
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occur, engaging in self-initiated symptom treatment strategies like reducing salt and fluid
intake or taking an extra diuretic, and evaluating the effectiveness of the implemented
treatments. Self-care confidence (i.e., self-efficacy related to the specific tasks of HF self-
care) is thought to be an important factor influencing the effectiveness of HF self-care (Cené
et al., 2013; Lee et al., 2011; Lee, Suwanno, & Riegel, 2009; Vellone et al., 2014). The
SCHFI version 6.2 directly reflects this naturalistic decision-making process and, thus, is
useful to investigators seeking to describe self-care and test the effectiveness of
interventions. In an earlier version (SCHFI v.4), when all three scales were added to yield a
single self-care score, Cronbach's alpha was 0.76 for the full scale and 0.56, 0.70, and 0.82
for the self-care maintenance, management, and confidence scales, respectively (Riegel et
al., 2004). When the SCHFI v.6.2 update was published, scale scores are no longer added
together. Considering the separate scales, Cronbach's alpha was reported to be 0.55 for self-
care maintenance, 0.60 self-care management, and 0.83 for self-care confidence (Riegel, et
al., 2009).Similarly poor alpha coefficients of the SCHFI have been reported by others (Kato
et al., 2013; Yu et al., 2011).
The data considered here refer to a sample of 549 subjects described in detail elsewhere
(Barbaranelli, Lee, Vellone, & Riegel, 2014). In these examples, confirmatory FAs were
conducted using Mplus 7.1 (Muthén & Muthén, 1998-2012). Due to some non-normality in
the data, parameter estimation was conducted using a robust maximum likelihood estimator
(i.e., Satorra-Bentler correction implemented via the estimator MLM).
As the first example we consider the analysis of the Self-Care Confidence scale. Riegel et al.
(2009) posited a single factor underlying the six items composing this scale. Accordingly,
we specified a one-factor model. This model showed an excellent fit: χ 2 (df = 9, N = 554) =
23.35, p < .01, Tucker-Lewis Index (TLI) = .99, Comparative Fit Index (CFI) = .99,
Standardized Root-Mean Square Residual (SRMR) = .023, and Root Mean Square Error of
Approximation (RMSEA) = .05, p = .37. Figure 1 presents the final fitted model: all factor
loadings were high, significant, and positive.
Reliability estimates for this model converged to the same value of 0.84 using both alpha
and the composite reliability (or omega) coefficients. This convergence is mainly due to the
substantial homogeneity of factor loadings. Although these loadings are not strictly equal,
they are almost uniformly high (see Raykov, 1997, in this regard).
The second analytical example refers to the Self-Care Maintenance Scale. Self-care
maintenance was hypothesized as an unidimensional scale by Riegel et al. (2009). However,
when testing a one-factor model with our data, the model proved to be largely
unsatisfactory, with the following poor-fit indices: χ 2 (df = 35, N = 549) =380, p < .001,
TLI = .41, CFI = .54, SRMR = .09, and RMSEA = .13, p < .001. Conversely, a four-factor
model adapted from Vellone, Riegel, and colleagues (2013), resulted in an excellent fit: χ 2
(df = 21, N = 549) = 49.19, p < .001, TLI = .93, CFI = .96, SRMR = .035 and RMSEA = .
049, p = .49. Figure 2 provides a graphical representation of this four-factor model. Factor
loadings were generally medium to high, thus attesting to a substantial proportion of
common variance among the items. Factor correlations were all positive and significant
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(except for one, see Figure 2), attesting to a significant and coherent association among the
different facets of self-care maintenance.
The self-care maintenance scale was intended to yield a single score, not four different
scores related to the different aspects of the construct. One item in the scale was problematic
and eliminated for this analysis, but even when the alpha coefficient was computed on the
nine items of the scale, a poor coefficient of .65 was obtained. Knowing that there are four
dimensions represented in this scale, more appropriate reliability coefficients that take into
account the multidimensionality of the scale are the global reliability index for
multidimensional scales (Raykov & Marcoulides, 2011), and the model-based internal
consistency coefficient (Bentler, 2009).
These coefficients were respectively .75 and .76 when derived from results in Figure 2.
Although the dimensionality of this scale is complex, as noted by Bentler (2006), “every
multidimensional coefficient implies a particular composite with maximal unidimensional
reliability” (p. 343). Thus, the final reliability estimates derived with appropriate methods
“can be interpreted to represent a unidimensional composite” (Bentler, 2006, p. 341).
What if Items Are Ordinal or Dichotomous?
Many of the scales used in nursing research are composed of items having an ordinal or
dichotomous response format. As a matter of fact, SCHFI items have a Likert-style graded
response format. One may question in this case whether the indices discussed in this article
adequately handle ordered categorical data. (Sijtsma and Van der Ark briefly refer to this
case on page x of their article.) Since ordinal and dichotomous response formats are not
uncommon in nursing research, we believe it is important to devote some extra space to this
issue. First, since the late 1970s, there has been concerted effort to develop estimation
methods suitable for conducting FA on ordinal and dichotomous data, especially by Muthén
(1984). More recently, new estimators have been developed and implemented in the Mplus
software. In particular, WLS-MV estimators are designed for use with ordinal or
dichotomous observed indicators of underlying continuous latent variables and have
performed well in sampling experiments using factor analysis models (Flora & Curran,
2004).
When the models in Figures 1 and 2 were analyzed, using these estimators, results overall
confirmed what is presented here, with estimates of factor loadings and of factor correlations
generally a little higher than those derived from robust maximum likelihood (ML)
(Barbaranelli et al., 2014). Raykov & Marcoulides (2011) advocate using internal coherence
coefficients, such as those discussed in this article, when items have less than five ordinal
responses options. The problem is that these methods for analyzing categorical ordinal
variables in FA consider these variables as discretization of underlying continuous variables.
Thus, when internal coherence estimates are computed from parameters derived from WLS-
MV, these estimates correspond to the continuous variables that underlie the observed
categorical responses—not the categorical responses themselves. The result is that methods
like WLS-MV tend to overestimate the value of internal consistency coefficients.
Conversely, when maximum likelihood methods are used in the analysis of ordinal or
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dichotomous data in FA, and parameters from this solution are used to compute reliability
estimates, reliability indices are generally underestimated. To our knowledge, the only
procedure that gives unbiased estimates of reliability coefficients derived from the analysis
of ordinal or dichotomous data is the one developed by Green and Yang (2009). When this
procedure was applied to our models using estimates derived from the WLS-MV method
(Barbaranelli et al., 2014), nonlinear structural equation model (SEM) reliability coefficients
of 0.84 and of 0.74 were obtained, thus substantially confirming the results obtained with
ML-robust estimators.
Concluding Remarks
We took advantage of this short article to expand upon issues raised by Sijtsma and Van der
Ark, and to provide practical examples of how reliability indices using the results of FA can
be applied to real-world data challenges that are common in nursing research. As we stated
throughout the article, dimensionality must be tested using FA before choosing an
appropriate measure of internal consistency. The two real data examples provided here
illustrate this process both in the case of a unidimensional scale (SCHFI Self-Care
Confidence), as well as a multidimensional scale (SCHFI Self-Care Maintenance). In both
cases, scale dimensionality was established by model fit, and internal consistency
coefficients developed from the results of FA generally outperformed values of Cronbach's
alpha. In fact, the reliability of the self-care maintenance scale was adequate only when a
coefficient was employed that reflects its multidimensional nature.
In closing, we want to draw your attention to Equation 8 of the Sijtsma and Van der Ark
article. This equation underlines the important issue that when row item scores are summed,
one is summing true-reliable scores with measurement error. Too often, applied researchers
and practitioners rely on the illusion that FA removes any sources of extraneous variability.
Sijtsma and Van der Ark also noted that factor models account for common variation of
indicators, but do not remove other sources of variability unless explicitly modeled. When
scale items are simply summed together to obtain a total scale, however, the scale will
contain variability that is due to common factor as well as that due to residual components
(DeShon, 2004). To limit, at least in part, the impact of such unwanted variance
components, factor scores can be used as an alternative to observed scores (Tabachnick &
Fidell, 2012). In computing factor scores, observed variables are not all treated the same;
instead, they are weighted by a coefficient that is proportional to the factor loading obtained
in the factor solution. We must acknowledge, however, that this practice may introduce
dependency from the particular sample on which these scaling coefficients were developed.
Consistent with this approach to scale construction, factor score determinacy coefficients
can be used (as an alternative to alpha) for evaluating the internal consistency of the factor
solution. Factor score determinacy represents the correlation between the estimated and true
factor scores; thus, it describes how well the factor is measured (Muthén & Muthén,
1998-2012). The larger the coefficient (e.g., ≥ .70 up to 1; Tabachnick & Fidell, 2007), the
better the factor is defined by the observed variables.
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Acknowledgments
The authors acknowledge this work was funded by the National Heart, Lung & Blood Institute (RO1 HL084394), by the Philadelphia Veterans Affairs Medical Center, VISN 4 Mental Illness Research, Education, and Clinical Center (MIREC), by the American Heart Association (11BGIA7840062), and by the U.S. Office of Research on Women's Health and National Institute of Child Health and Human Development (K12 HD043488-08). The content is solely the responsibility of the authors and does not necessarily represent the official views of the American Heart Association, Office of Research on Women's Health, or the National Institutes of Health.
The authors also gratefully acknowledge Sam Green for having applied his SAS macro for computing nonlinear structural equation model (SEM) reliability coefficients to their data.
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FIGURE 1. Confirmatory factor analysis model for the Self-Care Confidence Scale. N = 554. Adapted
from Barbaranelli et al. (2014) with permission from John Wiley and Sons.
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FIGURE 2. Confirmatory factor analysis model for the Self-Care Maintenance Scale. All factor
correlations are significant (p < .05) except the one with the asterisk. N = 549. Adapted from
Barbaranelli et al. (2014) with permission from John Wiley and Sons.
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